Chapter 3 Differential-Delay Equationspi.math.cornell.edu/~rand/randpdf/DDE-chapter3.pdfChapter 3 1...

Chapter 3 1

Differential-Delay Equations 2

Richard Rand 3

Abstract Periodic motions in DDE (Differential-Delay Equations) are typically 4

created in Hopf bifurcations. In this chapter we examine this process from several 5

points of view. Firstly we use Lindstedt’s perturbation method to derive the Hopf Bi- 6

furcation Formula, which determines the stability of the periodic motion. Then we 7

use the Two Variable Expansion Method (also known as Multiple Scales) to investi- 8

gate the transient behavior involved in the approach to the periodic motion. Next we 9

use Center Manifold Analysis to reduce the DDE from an an infinite dimensional 10

evolution equation on a function space to a two dimensional ODE (Ordinary Dif- 11

ferential Equation) on the center manifold, the latter being a surface tangent to the 12

eigenspace associated with the Hopf bifurcation. Finally we provide an application 13

to gene copying in which the delay is due to an observed time lag in the transcription 14

process. 15

3.1 Introduction 16

Some dynamical processes are modeled as differential-delay equations (abbreviated 17

DDE). An example is 18

dx(t)dt

=−x(t−T )− x(t)3 (3.1) 19

Here the rate of growth of x at time t is related both to the value of x at time t, and 20

also to the value of x at a previous time, t−T . 21

Applications of DDE include laser dynamics (Wirkus and Rand, 2002), where the 22

source of the delay is the time it takes light to travel from one point to another; 23

machine tool vibrations (Kalmar-Nagy et al., 2001), where the delay is due to the 24

Richard RandCornell University, Ithaca, NY 14853, USA,e-mail: [email protected]

from "Complex Systems: Fractionality, Time-delay and Synchronization" Springer 2011

86 Richard Rand

dependence of the cutting force on thickness of the rotating workpiece; gene dy- 1

namics (Verdugo and Rand, 2008a), where the delay is due to the time required for 2

messenger RNA to copy the genetic code and export it from the nucleus to the cyto- 3

plasm; investment analysis (Kot, 1979), where the delay is due to the time required 4

by bookkeepers to determine the current state of the system; and physiological dy- 5

namics (Camacho et al., 2004), where the delay comes from the time it takes a 6

substance to travel via the bloodstream from one organ to another. 7

A generalized version of Eq.(3.1) is 8

dx(t)dt

= αx(t)+βx(t−T )+ f (x(t),x(t−T )) (3.2) 9

where α and β are coefficients and f is a strictly nonlinear function of x(t) and 10

x(t−T ). Here the linear terms αx(t) and βx(t−T ) have been separated from the 11

strictly nonlinear terms, a step which facilitates stability analysis. 12

3.2 Stability of equilibrium 13

Equation (3.2) has the trivial equilibrium solution x(t) = 0. Is it stable? In order to 14

find out, we linearize Eq.(3.2) about x = 0: 15

dx(t)dt

= αx(t)+βx(t−T ). (3.3) 16

Since Eq.(3.3) has constant coefficients, we look for a solution in the form x = eλ t , 17

which gives the characteristic equation: 18

λ = α +βe−λT . (3.4) 19

Equation (3.4) is a transcendental equation and will in general have an infinite num- 20

ber of roots, which will either be real or will occur in complex conjugate pairs. The 21

equilibrium x = 0 will be stable if all the real parts of the roots are negative, and 22

unstable if any root has a positive real part. In the intermediate case in which no 23

roots have positive real part, but some roots have zero real part, the linear stability 24

analysis is inadequate, and nonlinear terms must be considered. 25

As an example, we consider Eq.(3.1), for which Eq.(3.4) becomes 26

λ =−e−λT . (3.5) 27

Since λ will be complex in general, we set λ = ν + iω , where ν and ω are the real 28

and imaginary parts. Substitution into Eq.(3.5) gives two real equations: 29

ν = −e−νT cosωT, (3.6) 30

ω = e−νT sinωT. (3.7) 31

3 Differential-Delay Equations 87

The question of stability will depend upon the value of the delay parameter T . Cer- 1

tainly when T = 0 the system is stable. By continuity, as T is increased from zero, 2

there will come a first positive value of T for which x = 0 is not (linearly) stable. 3

This can happen in one of two ways. Either a single real root will pass through the 4

origin in the complex-λ plane, or a pair of complex conjugate roots will cross the 5

imaginary axis. Since ν=ω=0 does not satisfy Eqs.(3.6),(3.7), the first case cannot 6

occur. 7

In order to consider the second case of a purely imaginary pair of roots, we set 8

ν = 0 in Eqs.(3.6),(3.7), giving 9

0 = −cosωT, (3.8) 10

ω = sinωT. (3.9) 11

Equation (3.8) gives ωT =π/2, whereupon Eq.(3.9) gives ω = 1, from which we 12

conclude that the critical value of delay T = Tcr=π/2. That is, x = 0 in Eq.(3.1) is 13

stable for T < π/2 and is unstable for T > π/2. Stability for T = Tcr=π/2 requires 14

consideration of nonlinear terms. 15

In order to check these results we numerically integrate Eq.(3.1) using the MAT- 16

LAB package DDE23. Note that this requires that the values of x be given on the 17

entire interval −T ≤ t ≤ 0. Figures 3.1 and 3.2 show the results of numerical in- 18

tegration using the initial condition x = 0.01 on −T ≤ t ≤ 0. Figure 3.1 is for 19

T = π/2−0.01 and shows stability, while Fig. 3.2 is for T = π/2+0.01 and shows 20

instability, in agreement with the foregoing analysis. 21

3.3 Lindstedt’s method 22

The change in stability observed in the preceding example will be accompanied by 23

the birth of a limit cycle in a Hopf bifurcation. In order to obtain an approximation 24

for the amplitude and frequency of the resulting periodic motion, we use Lindstedt’s 25

method (Rand and Armbruster, 1987; Rand, 2005). 26

We begin by stretching time, 27

τ = ωt. (3.10) 28

Replacing t by τ as independent variable, Eq.(3.1) may be written in the form: 29

ωdx(τ)

dτ=−x(τ−ωT )− x(τ)3. (3.11) 30

Next we choose the delay T to be close to the critical value Tcr=π/2: 31

T =π2

+∆ . (3.12) 32

We introduce a perturbation parameter ε ¿ 1 by scaling x: 33

88 Richard Rand

Fig. 3.1 Numerical integration of Eq.(3.1) for the initial condition x = 0.01 on −T ≤ t ≤ 0, forT = π/2−0.01.

Fig. 3.2 Numerical integration of Eq.(3.1) for the initial condition x = 0.01 on −T ≤ t ≤ 0, forT = π/2+0.01.


x =√

εu. (3.13) 1

Using Eq.(3.13), Eq.(3.11) becomes: 2

ωdu(τ)

dτ=−u(τ−ωT )− εu(τ)3. (3.14) 3

Next we scale ∆ 4

∆ = εµ , (3.15) 5

and we expand u and ω in power series of ε: 6

u(τ) = u0(τ)+ εu1(τ)+O(ε2), (3.16) 7

ω = 1+ εk1 +O(ε2), (3.17) 8

where we have used the fact that ω=1 when T =Tcr. 9

The delay term u(τ−ωT ) is handled by expanding it in Taylor series about ε=0: 10

u(τ−ωT ) = u(

τ− (1+ εk1 +O(ε2))(π2

+ εµ))

(3.18) 11

= u(

τ− π2− ε(k1

π2

+ µ)+O(ε2))

(3.19) 12

= u(τ− π2

)− ε(k1π2

+ µ)dudτ

(τ− π2

)+O(ε2). (3.20) 13

Substituting into Eq.(3.14) and collecting terms gives: 14

ε0 :du0

dτ+u0(τ− π

2) = 0, (3.21) 15

ε1 :du1

dτ+u1(τ− π

2) =−k1

du0

dτ+(k1

π2

+ µ)du0

dτ(τ− π

2)−u3

0. (3.22) 16

Since Eq.(3.1) is autonomous, we may choose the phase of the periodic motion 17

arbitrarily. This permits us to take the solution to Eq.(3.21) as: 18

u0(τ) = Acosτ, (3.23) 19

where A is the approximate amplitude of the periodic motion. Substituting Eq.(3.23) 20

into (3.22), we obtain: 21

du1

dτ+u1(τ− π

2) = k1Asinτ +(k1

π2

+ µ)Acosτ−A3(

34

cosτ +14

cos3τ)

,

(3.24) 22

For no secular terms, we equate to zero the coefficients of sinτ and cosτ on the 23

RHS of Eq.(3.24). This gives: 24

k1 = 0 and A =2√3√

µ. (3.25) 25

90 Richard Rand

Now A is the amplitude in u. In order to obtain the amplitude in x, we multiply the 1

second of Eqs.(3.25) by√

ε , which, together with Eqs.(3.13) and (3.15), gives 2

Amplitude of periodic motion in Eq.(3.1) =2√3

√∆ =

2√3

√(T − π

2

). (3.26) 3

This predicts, for example, that when T = π/2 + 0.01, the limit cycle born in the 4

Hopf will have approximate amplitude of 0.1155. For comparison, numerical inte- 5

gration gives a value of 0.1145, see Figure 3.3. 6

Fig. 3.3 Numerical integration of Eq.(3.1) for the initial condition x = 0.05 on −T ≤ t ≤ 0, forT = π/2+0.01.

3.4 Hopf bifurcation formula (Rand and Verdugo, 2007) 7

The treatment of the Hopf bifurcation in the previous section for Eq.(3.1) can be 8

generalized to apply to a wide class of DDEs. In this section we present a formula 9

for the amplitude of the resulting limit cycle for the DDE: 10

dxdt

= αx+βxd +a1x2 +a2xxd +a3x2d +b1x3 +b2x2xd +b3xx2

d +b4x3d , (3.27) 11


where x = x(t) and xd = x(t−T ). Here T is the delay. Associated with (3.27) is a 1

linear DDE 2

dxdt

= αx+βxd . (3.28) 3

We assume that (3.28) has a critical delay Tcr for which it exhibits a pair of pure 4

imaginary eigenvalues ±ωi corresponding to the solution 5

x = c1 cosωt + c2 sinωt (3.29) 6

Then for values of delay T which lie close to Tcr, 7

T = Tcr + µ, (3.30) 8

the nonlinear Eq.(3.27) may exhibit a periodic solution which can be written in the 9

approximate form: 10

x = Acosωt, (3.31) 11

where the amplitude A can be obtained from the following expression for A2: 12

A2 =PQ

µ, (3.32) 13

where 14

P = 4β 3 (β −α) (β +α)2 (−5β +4α) , (3.33) 15

Q = 15b4 β 6 Tcr +5b2 β 6 Tcr +3α b4 β 5 Tcr−15α b3 β 5 Tcr +α b2 β 5 Tcr 16

− 15α b1 β 5 Tcr−22a32 β 5 Tcr−7a2 a3 β 5 Tcr−14a1 a3 β 5 Tcr−3a2

2 β 5 Tcr 17

− 7a1 a2 β 5 Tcr−4a12 β 5 Tcr−12α2 b4 β 4 Tcr−3α2 b3 β 4 Tcr +6α2 b2 β 4 Tcr 18

− 3α2 b1 β 4 Tcr +12a32 α β 4 Tcr +37a2 a3 α β 4 Tcr +30a1 a3 α β 4 Tcr 19

+ 7a22 α β 4 Tcr +19a1 a2 α β 4 Tcr +18a1

2 α β 4 Tcr +12α3 b3 β 3 Tcr 20

+ 2α3 b2 β 3 Tcr +12α3 b1 β 3 Tcr +4a32 α2 β 3 Tcr−20a2 a3 α2 β 3 Tcr 21

− 16a1 a3 α2 β 3 Tcr−12a22 α2 β 3 Tcr−26a1 a2 α2 β 3 Tcr−8a1

2 α2 β 3 Tcr 22

− 8α4 b2 β 2 Tcr−4a2 a3 α3 β 2 Tcr +8a22 α3 β 2 Tcr +8a1 a2 α3 β 2 Tcr 23

+ 5b3 β 5 +15b1 β 5−15α b4 β 4 +α b3 β 4−15α b2 β 4 +3α b1 β 4−4a32 β 4

24

− 9a2 a3 β 4−18a1 a3 β 4−a22 β 4−9a1 a2 β 4−18a1

2 β 4−3α2 b4 β 325

+ 6α2 b3 β 3−3α2 b2 β 3−12α2 b1 β 3 +26a32 α β 3 +19a2 a3 α β 3

26

+ 30a1 a3 α β 3 +11a22 α β 3 + 33a1 a2 α β 3 +12a1

2 α β 3 +12α3 b4 β 227

+ 2α3 b3 β 2 +12α3 b2 β 2−8a32 α2 β 2−32a2 a3 α2 β 2−12a1 a3 α2 β 2

28

− 14a22 α2 β 2−18a1 a2 α2 β 2−8α4 b3 β −8a3

2 α3 β +8a2 a3 α3 β 29

+ 4a22 α3 β +8a2 a3 α4 (3.34) 30

92 Richard Rand

In Eq.(3.32), A is real so that A2 > 0, which means that µ must have the same sign 1

as PQ . 2

Eq.(3.34) depends on µ , α , β , ai, bi and Tcr. This equation may be alternately 3

written with Tcr expressed as a function of α and β . This relationship may be ob- 4

tained by considering the linear DDE (3.28). Substituting Eq.(3.31) into Eq.(3.28) 5

and equating to zero coefficients of sin(ωt) and cos(ωt), we obtain the two equa- 6

tions: 7

β sin(ω Tcr) =−ω, β cos(ω Tcr) =−α. (3.35) 8

Squaring and adding these we obtain 9

ω =√

β 2−α2. (3.36) 10

Substituting (3.36) into the second of (3.35), we obtain the desired relationship be- 11

tween Tcr and α and β : 12

Tcr =arccos

(−αβ

)√

β 2−α2. (3.37) 13

3.4.1 Example 1 14

As an example, we consider the following DDE: 15

dxdt

=−x−2xd − xxd − x3. (3.38) 16

This corresponds to the following parameter assignment in Eq.(3.27): 17

α =−1, β =−2, a1 = a3 = b2 = b3 = b4 = 0, a2 = b1 =−1. (3.39) 18

The associated linearized equation (3.28) is stable for zero delay. As the delay T 19

is increased, the origin first becomes unstable when T = Tcr, where Eq.(3.37) gives 20

that 21

Tcr =arccos −1

2√3

=2π

3√

3. (3.40) 22

Substituting (3.39) and (3.40) into (3.32), (3.33), (3.34), we obtain: 23

A2 =648µ

40√

3π +171= 1.667µ, (3.41) 24

where we have set 25

T = Tcr + µ =2π

3√

3+ µ = 1.2092+ µ. (3.42) 26


Thus the origin is stable for µ < 0 and unstable for µ > 0. In order for A2 in (3.41) to 1

be positive, we require that µ > 0. Therefore the limit cycle is born out of an unstable 2

equilibrium point. Since the stability of the limit cycle must be the opposite of the 3

stability of the equilibrium point from which it is born, we may conclude that the 4

limit cycle is stable and that we have a supercritical Hopf. This result is in agreement 5

with numerical integration of Eq.(3.38). 6

3.4.2 Derivation 7

In order to derive the result (3.32), (3.33), (3.34), we use Lindstedt’s method. We 8

begin by introducing a small parameter ε via the scaling 9

x = εu. (3.43) 10

The detuning µ of Eq.(3.30) is scaled like ε2: 11

T = Tcr + µ = Tcr + ε2µ. (3.44) 12

Next we stretch time by replacing the independent variable t by τ , where 13

τ = Ω t. (3.45) 14

This results in the following form of Eq.(3.27): 15

Ωdudτ

= αu+βud + ε(a1u2 +a2uud +a3u2d)+ ε2(b1u3 +b2u2ud +b3uu2

d +b4u3d),

(3.46) 16

where ud = u(τ−ΩT ). We expand Ω in a power series in ε , omitting the O(ε) term 17

for convenience, since it turns out to be zero: 18

Ω = ω + ε2k2 + · · · (3.47) 19

Next we expand the delay term ud : 20

ud = u(τ−ΩT ) = u(τ− (ω + ε2k2 + · · ·)(Tcr + ε2µ)) (3.48) 21

= u(τ−ωTcr− ε2(k2Tcr +ωµ)+ · · ·) (3.49) 22

= u(τ−ωTcr)− ε2(k2Tcr +ωµ)u′(τ−ωTcr)+O(ε3). (3.50) 23

Finally we expand u(τ) in a power series in ε: 24

u(τ) = u0(τ)+ εu1(τ)+ ε2u2(τ)+ · · · (3.51) 25

Substituting and collecting terms, we find: 26

94 Richard Rand

ωdu0

dτ−αu0(τ)−βu0(τ−ωTcr) = 0, (3.52) 1

ωdu1

dτ−αu1(τ)−βu1(τ−ωTcr) 2

= a1u0(τ)2 +a2u0(τ)u0(τ−ωTcr)+a3u0(τ−ωTcr)2, (3.53) 3

ωdu2

dτ−αu2(τ)−βu2(τ−ωTcr) = · · · (3.54) 4

where · · · stands for terms in u0 and u1, omitted here for brevity. We take the solution 5

of the u0 equation as (cf. Eq.(3.29) above): 6

u0(τ) = Acos(τ). (3.55) 7

We substitute (3.55) into (3.53) and obtain the following expression for u1: 8

u1(τ) = m1 sin(2τ)+m2 cos(2τ)+m3, (3.56) 9

where m1 is given by the equation: 10

m1 =− A2 (2a3 β +a2 β −2a1 β −2a3 α)√

β 2−α2

2β (β +α) (5β −4α). (3.57) 11

and where m2 and m3 are given by similar equations, omitted here for brevity. In 12

deriving (3.57), ω has been replaced by√

β 2−α2 as in Eq.(3.36). 13

Next the expressions for u0 and u1, Eqs.(3.55),(3.56), are substituted into the u2 14

equation, Eq.(3.54), and, after trigonometric simplifications have been performed, 15

the coefficients of the resonant terms, sinτ and cosτ , are equated to zero. This re- 16

sults in Eq.(3.32) for A2 as well as an expression for k2 (cf. Eq.(3.47)) which does 17

not concern us here. (Note that A = εA from Eqs.(3.31),(3.43),(3.55), and µ = ε2µ 18

from (3.44). The perturbation method gives A2 as a function of µ , but multiplication 19

by ε2 converts to a relation between A2 and µ .) 20

3.4.3 Example 2 21

As a second example, we consider the case in which the quantity Q in Eqs.(3.32),(3.34) 22

is zero. To generate such an example for the DDE (3.27), we embed the previous 23

example in a one-parameter family of DDE’s: 24

dxdt

=−x−2xd − xxd −λx3, (3.58) 25

and we choose λ so that Q = 0 in Eq.(3.32). This leads to the following critical value 26

of λ : 27

λ = λcr =4π +3

√3

18(2π +3√

3)= 0.0859 (3.59) 28


Since this choice for λ leads to Q = 0, Eq.(3.32) obviously cannot be used to find 1

the limit cycle amplitude A. Instead we use Lindstedt’s method, maintaining terms 2

of O(ε4). The correct scalings in this case turn out to be (cf.Eqs.(3.44),(3.47)): 3

T = Tcr + µ =2π

3√

3+ ε4µ, (3.60) 4

5

Ω = ω + ε2k2 + ε4k4 + · · · (3.61) 6

We find that the limit cycle amplitude A satisfies the equation: 7

A4 =−Γ µ , (3.62) 8

where we omit the closed form expression for Γ and give instead its approximate 9

value, Γ =620.477. 10

The analysis of this example has assumed that the parameter λ exactly takes on 11

the critical value given in Eq.(3.59). Let us consider a more robust model which 12

allows λ to be detuned: 13

λ = λcr +Λ =4π +3

√3

18(2π +3√

3)+ ε2Λ . (3.63) 14

Using Lindstedt’s method we obtain for this case the following equation on A: 15

A4 +σΛA2 +Γ µ = 0, (3.64) 16

where we omit the closed form expression for σ and give instead its approximate 17

value, σ=342.689. Eq.(3.64) can have 0,1, or 2 positive real roots for A, each of 18

which is a limit cycle in the original system. Thus the number of limit cycles which 19

are born in the Hopf bifurcation depends on the detuning coefficients Λ and µ . 20

Elementary use of the quadratic formula and the requirement that A2 be both real and 21

positive gives the following results: If µ < 0 then there is one limit cycle. If µ > 0 22

and σΛ < −2√

Γ µ then there are two limit cycles. If µ > 0 and σΛ > −2√

Γ µ 23

then there are no limit cycles. 24

3.4.4 Discussion 25

Although Lindstedt’s method is a formal perturbation method, i.e., lacking a proof 26

of convergence, our experience is that it gives the same results as the center mani- 27

fold approach, which has a rigorous mathematical foundation. The center manifold 28

approach has been described in many places, for example (Hassard et al., 1981; 29

Campbell et al., 1995; Stepan, 1989; Kalmar-Nagy et al., 2001; Rand, 2005), and 30

will be treated later in this Chapter. Since the DDE (3.27) is infinite dimensional 31

(for example the characteristic equation of the linear DDE (3.28) is transcendental 32

rather than polynomial, and hence has an infinite number of complex roots), the 33

96 Richard Rand

center manifold approach involves decomposing the original function space into a 1

two dimensional center manifold (in which the Hopf bifurcation takes place) and an 2

infinite dimensional function space representing the rest of the original phase space. 3

The center manifold procedure is much more complicated than the Hopf calcula- 4

tion. Stepan refers to the center manifold calculation as “long and tedious” ((Stepan, 5

1989), p.112), and Campbell et al. refer to it as “algebraically daunting” ((Campbell 6

et al., 1995), p.642). Thus the main advantage of the Hopf calculation is that it is 7

simpler to understand and easier to execute than the center manifold approach. 8

The idea of using Lindstedt’s method on bifurcation problems in DDE goes back 9

to a 1980 paper by Casal and Freedman (Casal and Freedman, 1980). That work 10

provided the algorithm but not the Hopf bifurcation formula. We present the general 11

expression for the Hopf bifurcation, as in Eqs.(3.32)—(3.34), as a convenience for 12

researchers in DDE. 13

3.5 Transient behavior 14

We have seen that Lindstedt’s method can be used to obtain an approximation for 15

the periodic motion of a DDE. This section is concerned with the use of perturbation 16

methods to obtain approximate expressions for the transient behavior of DDEs, e.g. 17

for the approach to a steady state periodic motion. In the case of ordinary differential 18

equations (ODEs), a very popular method for obtaining transient behavior is the 19

two variable expansion method (also know as multiple scales) (Cole, 1968; Nayfeh, 20

1973; Rand and Armbruster, 1987; Rand, 2005). In this section we show how this 21

method can be applied to a DDE. See also (Das and Chatterjee, 2002; Wang and Hu, 22

2003; Das and Chatterjee, 2005; Nayfeh, 2008). Although this approximate method 23

is strictly formal, its use is justified by center manifold considerations. Although 24

the DDE is an infinite dimensional system, a wide class of problems involves the 25

presence of a two dimensional invariant manifold, and it is the approximation of the 26

transient flow on this surface which is the goal of this perturbation method. 27

3.5.1 Example 28

In order to illustrate the manner in which this perturbation method may be applied 29

to DDEs, we choose a simple DDE problem, one that has an exact solution, namely: 30

dxdt

=−x(t−T ), T =π2

+ εµ . (3.65) 31


3.5.2 Exact solution 1

In Eq.(3.65) we set 2

x(t) = exp(λ t), (3.66) 3

giving the characteristic equation 4

λ =−exp(−λT ). (3.67) 5

When ε = 0, that is when T = π/2, Eq.(3.67) has the exact solution λ = i. Thus for 6

ε > 0 we set 7

λ = i+ ε(a+ ib). (3.68) 8

Substituting Eq.(3.68) into (3.67) and equating real and imaginary parts to zero, we 9

obtain the following two equations on a and b: 10

aε = exp(−aε2 µ− π aε

2

)sin

(bε2 µ + ε µ +

π bε2

), (3.69) 11

bε +1 = exp(−aε2 µ− π aε

2

)cos

(bε2 µ + ε µ +

π bε2

). (3.70) 12

If Eqs.(3.69),(3.70) were to be solved for a and b, we would obtain a solution to 13

(3.65) in the form: 14

x(t) = exp(εat)

sin (1+ εb)t,cos (1+ εb)t. (3.71) 15

In order to obtain a version of the exact solution (3.71) which will be useful for com- 16

paring solutions to Eq.(3.65) obtained by the perturbation method, we now derive 17

approximate expressions for a and b. Taylor expanding Eqs.(3.69),(3.70) for small 18

ε , we obtain 19

aε + · · ·= (bπ +2 µ) ε2

+ · · · (3.72) 20

1+bε + · · ·= 1− π aε2

+ · · · (3.73) 21

Solving Eqs.(3.72),(3.73) for a and b, we obtain the approximate expressions: 22

a =4 µ

π2 +4+O(ε), b =− 2π µ

π2 +4+O(ε) (3.74) 23

98 Richard Rand

3.5.3 Two variable expansion method (also known as multiple 1

scales) 2

In applying this method to the example of Eq.(3.65), we replace time t by two time 3

variables: regular time ξ = t, and slow time η = εt. The dependent variable x(t) is 4

then replaced by x(ξ ,η), and Eq.(3.65) becomes 5

∂x∂ξ

+ ε∂x∂η

=−x(ξ −T,η− εT ). (3.75) 6

Note that since T = π/2 + εµ , the delayed term may be expanded for small ε as 7

follows: 8

x(ξ −T,η− εT ) = x(ξ − π2

,η)− εµ∂xd

∂ξ− ε

π2

∂xd

∂η+O(ε2), (3.76) 9

where xd is an abbreviation for x(ξ − π2 ,η). Next we expand x = x0 + εx1 + O(ε2) 10

and collect terms in Eqs.(3.75),(3.76), giving 11

∂x0

∂ξ+ x0(ξ − π

2,η) = 0, (3.77) 12

13

∂x1

∂ξ+ x1(ξ − π

2,η) = µ

∂x0d

∂ξ+

π2

∂x0d

∂η− ∂x0

∂η. (3.78) 14

Eq.(3.77) has the periodic solution 15

x0 = R(η)cos(ξ −θ(η)). (3.79) 16

where as usual in this method, R(η) and θ(η) are as yet undetermined functions of 17

slow time η . The next step is to substitute Eq.(3.79) into (3.78). Before doing so, 18

we rewrite (3.78) by noting that (3.77) can be written in the form x0d =−∂x0/∂ξ : 19

∂x1

∂ξ+ x1(ξ − π

2,η) =−µ

∂ 2x0

∂ξ 2 −π2

∂ 2

∂η∂ξx0− ∂x0

∂η. (3.80) 20

Now we substitute (3.79) into (3.80) and require the coefficients of both cos(ξ −θ) 21

and sin(ξ −θ) to vanish, giving the following slow flow on R and θ : 22

R′+π2

Rθ ′−µR = 0, (3.81) 23

π2

R′−Rθ ′ = 0, (3.82) 24

where primes represent differentiation with respect to slow time η . Solving for R′ 25

and θ ′, we get 26


R′ =4µ

π2 +4R, (3.83) 1

θ ′ =2πµ

π2 +4, (3.84) 2

from which we obtain 3

R(η) = R0 exp(

4µηπ2 +4

), (3.85) 4

θ(η) =2πµ

π2 +4η +θ0. (3.86) 5

Eq.(3.79) thus gives 6

x≈ x0 = R0 exp(

4µεtπ2 +4

)cos

(t− 2πµ

π2 +4εt−θ0

), (3.87) 7

which agrees with the exact solution given by Eq.(3.71) with a and b given by (3.74). 8

3.5.4 Approach to limit cycle 9

Now let us use the two variable method on Eq.(3.1). We have seen by use of Lindst- 10

edt’s method that this DDE has a limit cycle of amplitude 2√µ/

√3, see Eq.(3.25). 11

The question arises as to the stability of this limit cycle. This may be determined as 12

follows: After scaling x as in Eq.(3.13), we may obtain Eq.(3.1) by adding the term 13

−εx(t)3 to the RHS of Eq.(3.65). This results in the term −x30 being added to the 14

RHS of Eqs.(3.78) and (3.80). After trigonometric reduction, this new term causes 15

the term −3R3/4 to be added to the RHS of Eq.(3.81), resulting in the new slow 16

flow 17

R′ =4µR−3R3

π2 +4, (3.88) 18

θ ′ =2πµ− (3π/2)R2

π2 +4. (3.89) 19

Here we see that Eq.(3.88) has two equilibria, R = 0 and R = 2√µ/

√3. Treat- 20

ing (3.88) as a flow on the R−line immediately shows that for µ > 0 the R = 0 21

equilibrium is unstable, a fact which we have already observed via a different ap- 22

proach, since R = 0 corresponds to the trivial solution of Eq.(3.1), which was inves- 23

tigated in Eqs.(3.5)–(3.9). In addition (3.88) shows that for µ > 0 the equilibrium 24

R = 2√µ/

√3 is stable, from which we may conclude that the corresponding limit 25

cycle is stable and that the Hopf bifurcation is supercritical. This conclusion agrees 26

with the numerical integration displayed in Figure 3.3. 27

100 Richard Rand

3.6 Center manifold analysis 1

We have seen earlier that the equilibrium solution x = 0 in Eq.(3.1) is stable for 2

T < π/2 and is unstable for T > π/2. The question remains as to the stability of 3

x = 0 when T = π/2. More generally, in order to determine the stability of the x=0 4

solution of a DDE in the form of Eq.(3.2), 5

dx(t)dt

= αx(t)+βx(t−T )+ f (x(t),x(t−T )), (3.90) 6

in the case that the delay T takes on its critical value Tcr, it is necessary to take 7

into account the effect of nonlinear terms. This may be accomplished by using a 8

center manifold reduction. In order to accomplish this, the DDE is reformulated 9

as an evolution equation on a function space. Our treatment closely follows that 10

presented in (Kalmar-Nagy et al., 2001). 11

The idea here is that the initial condition for Eq.(3.90) consists of a function de- 12

fined on −T ≤ t ≤ 0. As t increases from zero we may consider the piece of the 13

solution lying in the time interval [−T + t, t] as having evolved from the initial con- 14

dition function. In order to avoid confusion, the variable θ is used to identify a point 15

in the interval [−T,0], whereupon x(t + θ) will represent the piece of the solution 16

which has evolved from the initial condition function at time t. From the point of 17

view of the function space, t is a parameter, and it is θ which is the independent 18

variable. To emphasize this, we write: 19

xt(θ) = x(t +θ). (3.91) 20

The evolution equation, which acts on a function space consisting of continuously 21

differentiable functions on [−T,0], is written: 22

∂xt(θ)∂ t

=

∂xt(θ)∂θ

, for θ ∈ [−T,0),

αxt(0)+βxt(−T )+ f (xt(0),xt(−T )), for θ = 0.(3.92) 23

Here the DDE (3.90) appears as a boundary condition at θ = 0. The rest of the 24

interval goes along for the ride, which is to say that the equation ∂xt (θ)∂ t = ∂xt (θ)

∂θ is 25

an identity due to Eq.(3.91). 26

The RHS of Eq.(3.92) may be written as the sum of a linear operator A and a 27

nonlinear operator F : 28

Axt(θ) =

∂xt(θ)∂θ

for θ ∈ [−T,0),

αxt(0)+βxt(−T ) for θ = 0.(3.93) 29

Fxt(θ) =

0 for θ ∈ [−T,0),f (xt(0),xt(−T )) for θ = 0.

(3.94) 30


We now assume that the delay T is set at its critical value for a Hopf bifurcation, 1

i.e. the characteristic equation has a pair of pure imaginary roots, λ =±ωi. Corre- 2

sponding to these eigenvalues are a pair of eigenfunctions s1(θ) and s2(θ) which 3

satisfy the eigenequation: 4

A(s1(θ)+ is2(θ)) = iω(s1(θ)+ is2(θ)). (3.95) 5

That is, 6

As1(θ) = −ωs2(θ), (3.96) 7

As2(θ) = ωs1(θ). (3.97) 8

From the definition (3.93) of the linear operator A we find that s1(θ) and s2(θ) must 9

satisfy the following differential equations and boundary conditions: 10

ddθ

s1(θ) = −ωs2(θ), (3.98) 11

ddθ

s2(θ) = ωs1(θ), (3.99) 12

13

αs1(0)+β s1(−T ) = −ωs2(0), (3.100) 14

αs2(0)+β s2(−T ) = ωs1(0). (3.101) 15

Let’s illustrate this process by using Eq.(3.1) as an example. We saw earlier that at 16

T =Tcr=π/2, ω=1, which permits us to solve Eqs.(3.98), (3.99) as: 17

s1(θ) = c1 cosθ − c2 sinθ , (3.102) 18

s2(θ) = c1 sinθ + c2 cosθ , (3.103) 19

where c1 and c2 are arbitrary constants. In the case of Eq.(3.1), the boundary condi- 20

tions (3.100), (3.101) become (α=0, β=−1): 21

−s1(−π/2) = −s2(0), (3.104) 22

−s2(−π/2) = s1(0). (3.105) 23

Equations (3.102), (3.103) identically satisfy Eqs.(3.104), (3.105) so that the con- 24

stants of integration c1 and c2 remain arbitrary at this point. 25

In preparation for the center manifold analysis, we write the solution xt(θ) as 26

a sum of points lying in the center subspace (spanned by s1(θ) and s2(θ)) and of 27

points which do not lie in the center subspace, i.e., the rest of the solution, here 28

called w: 29

xt(θ) = y1(t)s1(θ)+ y2(t)s2(θ)+w(t)(θ). (3.106) 30

Here y1(t) and y2(t) are the components of xt(θ) lying in the directions s1(θ) and 31

s2(θ) respectively. 32

102 Richard Rand

The idea of the center manifold reduction is to find w as an approximate func- 1

tion of y1 and y2 (the center manifold), and then to substitute w(y1,y2) into 2

the equations on y1(t) and y2(t). The result is that we will have replaced the 3

original infinite dimensional system with a two dimensional approximation. 4

In order to find the equations on y1(t) and y2(t), we need to project xt(θ) onto the 5

center subspace. In this system, projections are accomplished by means of a bilinear 6

form: 7

(v,u) = v(0)u(0)+∫ 0

−Tv(θ +T )βu(θ)dθ , (3.107) 8

where u(θ) lies in the original function space, i.e. the space of continuously differ- 9

entiable functions defined on [−T,0], and where v(θ) lies in the adjoint function 10

space of continuously differentiable functions defined on [0,T ]. See the Appendix 11

to this chapter for a discussion of the adjoint operator A∗. 12

In order to project xt(θ) onto the center subspace, we will need the adjoint eigen- 13

functions n1(θ) and n2(θ) which satisfy the adjoint eigenequation: 14

A∗(n1(θ)+ in2(θ)) =−iω(n1(θ)+ in2(θ)), (3.108) 15

That is, 16

A∗n1(θ) = ωn2(θ), (3.109) 17

A∗n2(θ) = −ωn1(θ), (3.110) 18

where the adjoint operator A∗ is defined by 19

A∗v(θ) =

−dv(θ)dθ

for θ ∈ (0,T ],

αv(0)+βv(T ) for θ = 0.

(3.111) 20

In addition, the adjoint eigenfunctions ni are defined to be orthonormal to the eigen- 21

functions si: 22

(ni,s j) =

1, if i = j,0, if i 6= j. (3.112) 23

From the definition (3.111) of the linear operator A∗ we find that n1(θ) and n2(θ) 24

must satisfy the following differential equations and boundary conditions: 25

− ddθ

n1(θ) = ωn2(θ), (3.113) 26

− ddθ

n2(θ) = −ωn1(θ), (3.114) 27

αn1(0)+βn1(T ) = ωn2(0), (3.115) 28

αn2(0)+βn2(T ) = −ωn1(0). (3.116) 29

We continue to illustrate by using Eq.(3.1) as an example. With ω = 1, Eqs.(3.113), 30

(3.114) may be solved as: 31


n1(θ) = d1 cosθ −d2 sinθ , (3.117) 1

n2(θ) = d1 sinθ +d2 cosθ , (3.118) 2

where d1 and d2 are arbitrary constants. In the case of Eq.(3.1), the boundary con- 3

ditions (3.115), (3.116) become (α=0, β=−1): 4

−n1(π/2) = n2(0), (3.119) 5

−n2(π/2) = −n1(0). (3.120) 6

As in the case of the si eigenfunctions, Eqs.(3.117), (3.118) identically satisfy 7

Eqs.(3.119), (3.120) so that the constants of integration d1 and d2 remain arbitrary. 8

The four arbitrary constants c1, c2, d1, d2 of Eqs.(3.102),(3.103),(3.117),(3.118) 9

will be related by the orthonormality conditions (3.112). Let’s illustrate this by com- 10

puting (n1,s1) for the example of Eq.(3.1). Using the definition of the bilinear form 11

(3.107), we obtain: 12

(n1,s1) = n1(0)s1(0)+∫ 0

− π2

n1

(θ +

π2

)(−1)s1(θ)dθ , (3.121) 13

(n1,s1) =(2c2 +π c1) d2 +(2c1−π c2) d1

4= 1. (3.122) 14

Similarly, we find: 15

(n1,s2) =(π c2−2c1) d2 +(2c2 +π c1) d1

4= 0. (3.123) 16

The other two orthonormality conditions give no new information since it turns out 17

that (n2,s1) = −(n1,s2) and (n2,s2) = (n1,s1). Thus Eqs.(3.122) and (3.123) are 18

two equations in four unknowns, c1, c2, d1, d2. Without loss of generality we take 19

d1 = 1 and d2 = 0, giving c1 = 8π2+4 ,c2 =− 4π

π2+4 . Thus the eigenfunctions si and ni 20

for Eq.(3.1) become: 21

s1(θ) =4π sinθ +8 cosθ

π2 +4, (3.124) 22

s2(θ) =8 sinθ −4π cosθ

π2 +4, (3.125) 23

n1(θ) = cosθ , (3.126) 24

n2(θ) = sinθ . (3.127) 25

Recall that our purpose in solving for n1 and n2 was to obtain equations on y1(t) and 26

y2(t), the components of xt(θ) lying in the directions s1(θ) and s2(θ) respectively, 27

see Eq.(3.106). We have: 28

y1(t) = (n1,xt), y2(t) = (n2,xt). (3.128) 29

Differentiating (3.128) with respect to t: 30

104 Richard Rand

y1(t) = (n1, xt), y2(t) = (n2, xt). (3.129) 1

Let us consider the first of these: 2

y1(t) = (n1, xt) = (n1,Axt +Fxt) = (n1,Axt)+(n1,Fxt). (3.130) 3

Now by definition of the adjoint operator A∗, 4

(n1,Axt) = (A∗n1,xt) = (ωn2,xt) = ω(n2,xt) = ωy2. (3.131) 5

So we have 6

y1 = ωy2 +(n1,Fxt), 7

and similarly (3.132) 8

y2 = −ωy1 +(n2,Fxt). 9

In Eqs.(3.132), the quantities (ni,Fxt) are given by (cf. Eq.(3.107)): 10

(ni,Fxt) = ni(0)Fxt(0)+∫ 0

−Tni(θ +T )βFxt(θ)dθ (3.133) 11

= ni(0) f (xt(0),xt(−T )) since Fxt(θ)=0 unless θ=0, (3.134) 12

in which xt = y1(t)s1(θ) + y2(t)s2(θ) + w(t)(θ). Continuing with the example of 13

Eq.(3.1), Eqs.(3.132) become, using f =−x(t)3: 14

y1 = y2−(

8y1

π2 +4− 4πy2

π2 +4+w(θ = 0)

)3

and y2 =−y1, (3.135) 15

where we have used s1(0)= 8π2+4 , s2(0)= −4π

π2+4 , n1(0)=1 and n2(0)=0 from Eqs.(3.124) 16

–(3.127). 17

The next step is to look for an approximate expression for the center manifold, 18

which is tangent to the y1-y2 plane at the origin, and which may be written in the 19

form: 20

w(y1,y2)(θ) = m1(θ)y21 +m2(θ)y1y2 +m3(θ)y2

2. (3.136) 21

The procedure is to substitute (3.136) into the equations of motion, collect terms, 22

and solve for the unknown functions mi(θ). Then the resulting expression is to be 23

substituted into the y1-y2 equations (3.132). Note that if this is done for the example 24

of Eq.(3.1), i.e. for Eqs.(3.135), the contribution made by w will be of degree 4 25

and higher in the yi. However, stability of the origin will be determined by terms 26

of degree 2 and 3, according to the following formula (obtainable by averaging). 27

Suppose the y1-y2 equations are of the form: 28

y1 = ωy2 +h(y1,y2) and y2 =−ωy1 +g(y1,y2). (3.137) 29

Then the stability of the origin is determined by the sign of the quantity Q, where 30


16Q = h111 +h122 +g112 +g222 1

− 1ω

[h12(h11 +h22)−g12(g11 +g22)−h11g11 +h22g22] , (3.138) 2

where subscripts represent partial derivatives, which are to be evaluated at y1 = y2 = 3

0.Q > 0 means unstable, Q < 0 means stable. See ( Guckenheimer and Holmes, 4

1983) pp.154-156, where it is shown that the flow on the y1-y2 plane in the neigh- 5

borhood of the origin can be approximately written in polar coordinates as: 6

drdt

= Qr3 +O(r5). (3.139) 7

Applying this criterion to Eqs.(3.135) (in which w is assumed to be of the form 8

(3.136) and hence contributes terms of higher order in yi), we find: 9

Q =− 48

(π2 +4)2 =−0.2495. (3.140) 10

The origin in Eq.(3.1) for T = Tcr = π/2 is therefore predicted to be stable. This 11

result is in agreement with numerical integration, see Fig. 3.4. 12

Fig. 3.4 Numerical integration of Eq.(3.1) for the initial condition x = 0.1 on −T ≤ t ≤ 0, forT = π/2.

106 Richard Rand

Now let’s change the example a little so that w plays a significant role in determining 1

the stability of the origin: 2

dx(t)dt

=−x(t−T )− x(t)2. (3.141) 3

Since the linear parts of this example and of the previous example are the same, our 4

previously derived expressions for s1, s2, n1 and n2, Eqs.(3.124)–(3.127), still apply. 5

Eqs.(3.135) now become: 6

y1 = y2−(

8y1

π2 +4− 4πy2

π2 +4+w(θ = 0)

)2

and y2 =−y1. (3.142) 7

So our goal now is to find the functions mi(θ) in the expression for the center man- 8

ifold (3.136), and then to substitute the result into Eq.(3.142) and use the formula 9

(3.138) to determine stability. 10

We begin by differentiating the expression for the center manifold (3.136) with 11

respect to t: 12

∂w(y1,y2)(θ)∂ t

= 2m1(θ)y1y1 +m2(θ)(y1y2 + y2y1)+2m3(θ)y2y2. (3.143) 13

We substitute the equations (3.132) on y1 and y2 in (3.143) and neglect terms of 14

degree higher than 2 in the yi: 15

∂w(y1,y2)(θ)∂ t

= 2m1(θ)y1ωy2 +m2(θ)(−y1ωy1 + y2ωy2) 16

−2m3(θ)y2ωy1 + · · · (3.144) 17

= ω[−m2(θ)y21 +2(m1(θ)−m3(θ))y1y2 18

+m2(θ)y22]+ · · · (3.145) 19

We obtain conditions on the functions mi(θ) by deriving another expression for w 20

and equating them. Let us differentiate Eq.(3.106) with respect to t: 21

∂xt(θ)∂ t

= y1(t)s1(θ)+ y2(t)s2(θ)+∂w(t)(θ)

∂ t. (3.146) 22

Using the functional DE (3.92)–(3.94), and rearranging terms, we get: 23


∂w(t)(θ)∂ t

=∂xt(θ)

∂ t− y1(t)s1(θ)− y2(t)s2(θ) (3.147) 1

= Axt(θ)+Fxt(θ)− y1(t)s1(θ)− y2(t)s2(θ) (3.148) 2

= A(y1(t)s1(θ)+ y2(t)s2(θ)+w(t)(θ)) 3

+Fxt(θ)− y1(t)s1(θ)− y2(t)s2(θ) (3.149) 4

= y1As1 + y2As2 +Aw+Fxt − y1s1− y2s2 (3.150) 5

= y1(−ωs2)+ y2(ωs1)+Aw+Fxt 6

−(ωy2 +(n1,Fxt))s1− (−ωy1 +(n2,Fxt))s2 (3.151) 7

= Aw+Fxt − (n1,Fxt)s1− (n2,Fxt)s2 (3.152) 8

where we have used Eqs.(3.96), (3.97) and (3.132), and where the quantities 9

(ni,Fxt) are given by Eq.(3.134). 10

Eq.(3.152) is an equation for the time evolution of w. Since the operator A is 11

defined differently for θ ∈ [−T,0) and for θ = 0, we consider each of these cases 12

separately when we substitute Eq.(3.136) for the center manifold. In the θ ∈ [−T,0) 13

case, Eq.(3.152) becomes: 14

∂w(t)(θ)∂ t

= m′1y2

1 +m′2y1y2 +m′

3y22− (n1,Fxt)s1(θ)− (n2,Fxt)s2(θ), (3.153) 15

where primes denote differentiation with respect to θ . In the θ = 0 case, Eq.(3.152) 16

becomes: 17

∂w(t)(θ)∂ t

= α(m1(0)y21 +m2(0)y1y2 +m3(0)y2

2) 18

+β (m1(−T )y21 +m2(−T )y1y2 +m3(−T )y2

2) 19

+ f (xt(0),xt(−T ))− (n1,Fxt)s1(0)− (n2,Fxt)s2(0). (3.154) 20

Now we equate Eqs.(3.153) and (3.154) to the previously derived expression for w, 21

Eq.(3.145). Equating (3.153) to (3.145), we get: 22

m′1y2

1 +m′2y1y2 +m′

3y22− (n1,Fxt)s1(θ)− (n2,Fxt)s2(θ) = 23

ω[−m2y21 +2(m1−m3)y1y2 +m2y2

2]+ · · · (3.155) 24

Equating (3.154) to (3.145), we get: 25

α(m1(0)y21 +m2(0)y1y2 +m3(0)y2

2) 26

+β (m1(−T )y21 +m2(−T )y1y2 +m3(−T )y2

2) 27

+ f (xt(0),xt(−T ))− (n1,Fxt)s1(0)− (n2,Fxt)s2(0) = 28

ω[−m2(0)y21 +2(m1(0)−m3(0))y1y2 +m2(0)y2

2]+ · · · (3.156) 29

Now we equate coefficients of y21, y1y2 and y2

2 in Eqs.(3.155) and (3.156) and so 30

obtain 3 first order ODE’s on m1, m2 and m3 and 3 boundary conditions. From 31

Eq.(3.134), the nonlinear terms (ni,Fxt) become: 32

108 Richard Rand

(ni,Fxt) = ni(0) f (xt(0),xt(−T )), (3.157) 1

in which xt = y1(t)s1(θ)+ y2(t)s2(θ)+ w(t)(θ) ≈ y1(t)s1(θ)+ y2(t)s2(θ). In the 2

case of the example system (3.141) we have α = 0, β = −1, ω = 1, T = π/2, 3

f (x(t),x(t−T )) =−x(t)2 and 4

f (xt(0),xt(−T )) =−xt(0)2 =−(y1s1(0)+ y2s2(0))2. (3.158) 5

For this example, Eq.(3.155) becomes 6

m′1y2

1 +m′2y1y2 +m′

3y22 +(y1s1(0)+ y2s2(0))2s1(θ) 7

=−m2y21 +2(m1−m3)y1y2 +m2y2

2 (3.159) 8

which gives the following 3 ODE’s on mi(θ): 9

m′1 + s1(0)2s1(θ) = −m2, (3.160) 10

m′2 +2s1(0)s2(0)s1(θ) = 2(m1−m3), (3.161) 11

m′3 + s2(0)2s1(θ) = m2. (3.162) 12

For this example, Eq.(3.156) becomes 13

−(m1(−π/2)y21 +m2(−π/2)y1y2 +m3(−π/2)y2

2) 14

−(y1s1(0)+ y2s2(0))2 +(y1s1(0)+ y2s2(0))2s1(0) = 15

−m2(0)y21 +2(m1(0)−m3(0))y1y2 +m2(0)y2

2, (3.163) 16

which gives the following 3 boundary conditions on mi(θ): 17

−m1(−π/2)− s1(0)2 + s1(0)3 = −m2(0), (3.164) 18

−m2(−π/2)−2s1(0)s2(0)+2s1(0)2s2(0) = 2(m1(0)−m3(0)), (3.165) 19

−m3(−π/2)− s2(0)2 + s2(0)2s1(0) = m2(0). (3.166) 20

So we have 3 linear ODE’s (3.160)–(3.162) with 3 boundary conditions (3.164)– 21

(3.166) for the mi(θ). In these equations, s1 and s2 are given by Eqs.(3.124), (3.125). 22

The solution of these equations is algebraically complicated. I used a computer al- 23

gebra system to obtain a closed form solution for the mi(θ). For brevity, a numerical 24

version of the center manifold is given below: 25

w = 26

(0.20216 sin2θ +0.16022 cos2θ −0.6953 sinθ +0.39537 cosθ −0.5768) y12

27

+(0.32044 sin2θ −0.40432 cos2θ +0.09393 sinθ +0.5034 cosθ) y1 y2 28

+(−0.20216 sin2θ −0.16022 cos2θ +0.0299 sinθ +0.64984 cosθ −0.5768) y22. 29

(3.167) 30


Next we substitute the algebraic version of Eq.(3.167) into the y1-y2 Eqs.(3.142) and 1

use Eq.(3.138) to compute the stability parameter Q: 2

Q =32 (9−π)

5 (π2 +4)2 = 0.19491 > 0. (3.168) 3

Thus the center manifold analysis predicts that origin of Eq.(3.141) is unstable. This 4

result is in agreement with numerical integration, see Fig. 3.5. 5

Fig. 3.5 Numerical integration of Eq.(3.141) for the initial condition x = 0.04 on −T ≤ t ≤ 0, forT = π/2.

3.6.1 Appendix: The adjoint operator A∗ 6

The adjoint operator A∗ is defined by the relation: 7

(v,Au) = (A∗v,u), (3.169) 8

where the bilinear form (v,u) is given by Eq.(3.107): 9

(v,u) = v(0)u(0)+∫ 0

−Tv(θ +T )βu(θ)dθ , (3.170) 10

110 Richard Rand

where u(θ) lies in the original function space, i.e. the space of continuously differ- 1

entiable functions defined on [−T,0], and where v(θ) lies in the adjoint function 2

space of continuously differentiable functions defined on [0,T ]. 3

4

The linear operator A is given by Eq.(3.93): 5

Au(θ) =

du(θ)dθ

for θ ∈ [−T,0),

αu(0)+βu(−T ) for θ = 0,(3.171) 6

from which (v,Au) can be written as follows: 7

(v,Au) = v(0)Au(0)+∫ 0

−Tv(θ +T )βAu(θ)dθ (3.172) 8

= v(0)[αu(0)+βu(−T )]+∫ 0

−Tv(θ +T )β

du(θ)dθ

dθ . (3.173) 9

Using integration by parts, the last term of Eq.(3.173) can be written: 10

∫ 0

−Tv(θ +T )β

du(θ)dθ

dθ 11

= v(θ +T )βu(θ)| 0−T −

∫ 0

−Tβu(θ)

dv(θ +T )dθ

dθ (3.174) 12

= v(T )βu(0)− v(0)βu(−T )−∫ T

φ=0βu(φ −T )

dv(φ)dφ

dφ (3.175) 13

where φ = θ +T . Substituting (3.175) into (3.173), we get 14

(v,Au) = [αv(0)+βv(T )]u(0)+∫ T

φ=0

(−dv(φ)

dφ

)βu(φ −T )dφ (3.176) 15

= (A∗v,u) (3.177) 16

from which we may conclude that the adjoint operator A∗ is given by: 17

A∗v(φ) =

−dv(φ)dφ

for φ ∈ (0,T ],

αv(0)+βv(T ) for φ = 0.(3.178) 18

3.7 Application to gene expression (Verdugo and Rand, 2008a) 19

This section offers a timely example showing how DDEs occur in a mathemati- 20

cal model of gene expression (Monk, 2003). The biology of the problem may be 21

described as follows: A gene, i.e. a section of the DNA molecule, is copied (tran- 22


scribed) onto messenger RNA (mRNA), which diffuses out of the nucleus of the 1

cell into the cytoplasm, where it enters a subcellular structure called a ribosome. In 2

the ribosome the genetic code on the mRNA produces a protein (a process called 3

translation). The protein then diffuses back into the nucleus where it represses the 4

transcription of its own gene. 5

Dynamically speaking, this process may result in a steady state equilibrium, in 6

which the concentrations of mRNA and protein are constant, or it may result in an 7

oscillation. In this section we analyze a simple model previously proposed in the 8

biological literature (Monk, 2003), and we show that the transition between equilib- 9

rium and oscillation is a Hopf bifurcation. The model takes the form of two equa- 10

tions, one an ordinary differential equation (ODE) and the other a delayed differen- 11

tial equation (DDE). The delay is due to an observed time lag in the transcription 12

process. 13

Oscillations in biological systems with delay have been dealt with previously in 14

(Mahaffy, 1988; Mahaffy et al., 1992; Mocek et al., 2005). 15

The model equations investigated here involve the variables M(t), the concentration 16

of mRNA, and P(t), the concentration of the associated protein (Monk, 2003): 17

M = αm

(1

1+(

PdP0

)n

)− µm M (3.179) 18

P = αp M−µp P (3.180) 19

where dots represent differentiation with respect to time t, and where we use the 20

subscript d to denote a variable which is delayed by time T , thus Pd = P(t−T ). The 21

model constants are as given in (Monk, 2003): αm is the rate at which mRNA is tran- 22

scribed in the absence of the associated protein, αp is the rate at which the protein 23

is produced from mRNA in the ribosome, µm and µp are the rates of degradation of 24

mRNA and of protein, respectively, P0 is a reference concentration of protein, and n 25

is a parameter. We assume µm=µp=µ . 26

3.7.1 Stability of equilibrium 27

We begin by rescaling Eqs. (3.179) and (3.180). We set m = Mαm

, p = Pαmαp

, and 28

p0 = P0αmαp

, giving: 29

m =1

1+(

pdp0

)n −µm, (3.181) 30

p = m−µ p. (3.182) 31

Equilibrium points, (m∗, p∗), for (3.181) and (3.182) are found by setting m = 0 and 32

p = 0 33

112 Richard Rand

µm∗ =1

1+(

p∗p0

)n (3.183) 1

m∗ = µ p∗ (3.184) 2

Eliminating m∗ from Eqs. (3.183) and (3.184), we obtain an equation on p∗: 3

(p∗)n+1 + pn0 p∗ − pn

0µ2 = 0. (3.185) 4

Next we define ξ and η to be deviations from equilibrium: ξ = ξ (t) = m(t)−m∗, 5

η = η(t) = p(t)− p∗, and ηd = η(t−T ). This results in the nonlinear system: 6

ξ =1

1+(

ηd+p∗p0

)n −µ(m∗+ξ ), (3.186) 7

η = ξ −µη . (3.187) 8

Expanding for small values of ηd , Eq.(3.186) becomes: 9

ξ = −µξ −K ηd +H2 η2d +H3 η3

d + · · · (3.188) 10

where K, H2 and H3 depend on p∗, p0, and n as follows: 11

K =nβ

p∗(1+β )2 , where β =

(p∗

p0

)n

, (3.189) 12

H2 =β n (β n−n+β +1)

2 (β +1)3 p∗2, (3.190) 13

H3 = −β n(β 2 n2−4β n2 +n2 +3β 2 n−3n+2β 2 +4β +2

)

6 (β +1)4 p∗3. (3.191) 14

Next we analyze the linearized system coming from Eqs. (3.188) and (3.187): 15

ξ = −µ ξ −K ηd , (3.192) 16

η = ξ −µ η . (3.193) 17

Stability analysis of Eqs. (3.192) and (3.193) shows that for T = 0 (no delay), the 18

equilibrium point (m∗, p∗) is a stable spiral. Increasing the delay, T , in the linear 19

system (3.192)–(3.193), will yield a critical delay, Tcr, such that for T > Tcr, (m∗, p∗) 20

will be unstable, giving rise to a Hopf bifurcation. For T = Tcr the system (3.192)– 21

(3.193) will exhibit a pair of pure imaginary eigenvalues ±ωi corresponding to the 22

solution 23

ξ (t) = Bcos(ωt +φ), (3.194) 24

η(t) = Acosωt, (3.195) 25


where A and B are the amplitudes of the η(t) and ξ (t) oscillations, and where φ is 1

a phase angle. Note that we have chosen the phase of η(t) to be zero without loss 2

of generality. Then for values of delay T close to Tcr, 3

T = Tcr +∆ , (3.196) 4

the nonlinear system (3.181)–(3.182) is expected to exhibit a periodic solution (a 5

limit cycle) which can be written in the approximate form of Eqs. (3.194), (3.195). 6

Substituting Eqs. (3.194) and (3.195) into Eqs. (3.192) and (3.193) and solving for 7

ω and Tcr we obtain 8

ω =√

K−µ2, (3.197) 9

Tcr =arctan

(2µ√

K−µ2

K−2µ2

)√

K−µ2. (3.198) 10

3.7.2 Lindstedt’s method 11

We use Lindstedt’s Method (Rand and Verdugo, 2007) on Eqs. (3.188) and (3.187). 12

We begin by changing the first order system into a second order DDE. This results 13

in the following form: 14

η +2 µ η + µ2 η =−K ηd +H2 η2d +H3 η3

d + · · · (3.199) 15

where K, H2 and H3 are defined by Eqs. (3.189)–(3.191). We introduce a small 16

parameter ε via the scaling 17

η = εu. (3.200) 18

The detuning ∆ of Eq. (3.196) is scaled like ε2, ∆ = ε2δ : 19

T = Tcr +∆ = Tcr + ε2δ . (3.201) 20

Next we stretch time by replacing the independent variable t by τ , where 21

τ = Ω t. (3.202) 22

This results in the following form of Eq. (3.199): 23

Ω 2 d2udτ2 + 2 µ Ω

dudτ

+ µ2 u = −K ud + ε H2 u2d + ε2 H3 u3

d , (3.203) 24

where ud = u(τ−Ω T ). We expand Ω in a power series in ε , omitting the O(ε) for 25

convenience, since it turns out to be zero: 26

Ω = ω + ε2k2 + · · · (3.204) 27

114 Richard Rand

Next we expand the delay term ud : 1

ud = u(τ−ΩT ) = u(τ− (ω + ε2k2 + · · ·)(Tcr + ε2δ )) (3.205) 2

= u(τ−ωTcr− ε2(k2Tcr +ωδ )+ · · ·) (3.206) 3

= u(τ−ωTcr)− ε2(k2Tcr +ωδ )u′(τ−ωTcr)+O(ε3) (3.207) 4

Now we expand u(τ) in a power series in ε: 5

u(τ) = u0(τ)+ εu1(τ)+ ε2u2(τ)+ · · · (3.208) 6

Substituting and collecting terms, we find: 7

ω2 d2u0

dτ2 +2µωdu0

dτ+Ku0(τ−ωTcr)+ µ2u0 = 0 (3.209) 8

ω2 d2u1

dτ2 +2µωdu1

dτ+Ku1(τ−ωTcr)+ µ2u1 = H2u2

0(τ−ωTcr) (3.210) 9

ω2 d2u2

dτ2 +2µωdu2

dτ+Ku2(τ−ωTcr)+ µ2u2 = · · · (3.211) 10

where · · · stands for terms in u0 and u1, omitted here for brevity. We take the solution 11

of the u0 equation as: 12

u0(τ) = Acosτ, (3.212) 13

where from Eqs. (3.195) and (3.200) we know A = Aε . Next we substitute (3.212) 14

into (3.38) and obtain the following expression for u1: 15

u1(τ) = m1 sin2τ +m2 cos2τ +m3, (3.213) 16

where m1 is given by the equation: 17

m1 =−2 A2 H2 µ√

K−µ2(µ2−K

) (2 µ2−3K

)

K (16 µ6−39K µ4 +18K2 µ2 +9K3), (3.214) 18

and where m2 and m3 are given by similar equations, omitted here for brevity. We 19

substitute Eqs. (3.212) and (3.213) into (3.40), and after trigonometric simplifica- 20

tions have been performed, we equate to zero the coefficients of the resonant terms 21

sinτ and cosτ . This yields the amplitude, A, of the limit cycle that was born in the 22

Hopf bifurcation: 23

A2 =PQ

∆ , (3.215) 24

where 25

P = −8K2 (µ4−K2)(16 µ6−39K µ4 +18K2 µ2 +9K3), (3.216) 26

Q = Q0 Tcr + Q1, (3.217) 27

and 28


Q0 = 48H3 K2 µ8 +16H22 K µ8−69H3 K3 µ6 +32H2

2 K2 µ61

− 63H3 K4 µ4−162H22 K3 µ4 +81H3 K5 µ2 +108H2

2 K4 µ22

+ 27H3 K6 +30H22 K5, (3.218) 3

Q1 = 96H3 K µ9 +64H22 µ9−138H3 K2 µ7−16H2

2 K µ74

− 126H3 K3 µ5−308H22 K2 µ5 +162H3 K4 µ3 +296H2

2 K3 µ35

+ 54H3 K5 µ +12H22 K4 µ. (3.219) 6

Eq. (3.217) depends on µ , K, H2, H3, and Tcr. By using Eq. (3.198) we may express 7

Eq. (3.217) as a function of µ , K, H2, and H3 only. Removal of secular terms also 8

yields a value for the frequency shift k2, where, from Eq. (3.204), we have Ω = 9

ω + ε2k2: 10

k2 =−RQ

δ . (3.220) 11

where Q is given by (3.217) and 12

R =√

K−µ2 Q0. (3.221) 13

An expression for the amplitude B of the periodic solution for ξ (t) (see Eq. (3.194)) 14

may be obtained directly from Eq. (3.187) by writing ξ = η + µη , where η ∼ 15

Acosωt. We find: 16

B =√

KA, (3.222) 17

where K and A are given as in (3.189) and (3.215) respectively. 18

3.7.3 Numerical example 19

Using the same parameter values as in (Monk, 2003) 20

µ = 0.03/min , p0 = 100 , n = 5, (3.223) 21

we obtain 22

p∗ = 145.9158 , m∗ = 4.3774, (3.224) 23

24

K = 3.9089×10−3 , H2 = 6.2778×10−5 , H3 =−6.4101×10−7, (3.225) 25

26

Tcr = 18.2470 , w = 5.4854×10−2 ,2πw

= 114.5432. (3.226) 27

Here the delay Tcr and the response period 2π/ω are given in minutes. Substituting 28

(3.224)–(3.226) into (3.215)–(3.222) yields the following equations: 29

A = 27.0215√

∆ , (3.227) 30

k2 = −2.4512×10−3 δ , (3.228) 31

B = 1.6894√

∆ . (3.229) 32

116 Richard Rand

Note that since Eq. (3.227) requires ∆ > 0 for the limit cycle to exist, and since we 1

saw in Eqs. (3.192) and (3.193) that the origin is unstable for T > Tcr, i.e. for ∆ > 0, 2

we may conclude that the Hopf bifurcation is supercritical, i.e., the limit cycle is 3

stable. 4

5

Multiplying (3.228) by ε2 and substituting into (3.204) we obtain: 6

Ω = 5.4854×10−2 − 2.4512×10−3 ∆ (3.230) 7

where ∆ = T − Tcr = T − 18.2470. Plotting the period, 2πΩ , against the delay, T , 8

yields the graph shown in Figure 3.6. These results are in agreement with those 9

obtained by numerical integration of the original Eqs. (3.179) and (3.180) and with 10

those presented in (Monk, 2003). 11

Fig. 3.6 Period of oscillation, 2πΩ , plotted as a function of delay T , where Ω is given by Eq.(3.230).

The initiation of oscillation at T = Tcr = 18.2470 is due to a supercritical Hopf bifurcation, and ismarked in the Figure with a dot.

3.8 Exercises 12

Exercise 1 13

For which values of the delay T > 0 is the trivial solution in the following DDE 14

stable? 15


dx(t)dt

= x(t)−2x(t−T ). (3.231) 1

Exercise 2 2

Use Lindstedt’s method to find an approximation for the amplitude of the limit cycle 3

in the following DDE: 4

dx(t)dt

=−x(t−T )+ x(t−T )3. (3.232) 5

Exercise 3 6

Use the center manifold approach to determine the stability of the x=0 solution in 7

the following DDE: 8

dx(t)dt

=−x(t− π2

)(1+ x(t)). (3.233) 9

Here is an outline of the steps involved in this complicated calculation: 10

1. Show that the parameters of the linearized equation 11

dx(t)dt

=−x(

t− π2

), (3.234) 12

have been chosen so that the delay is set at its critical value for a Hopf bifurcation, 13

i.e. the characteristic equation has a pair of pure imaginary roots, λ =±ωi. Find 14

ω . 15

2. Find the eigenfunctions s1(θ), s2(θ) and the adjoint eigenfunctions n1(θ), n2(θ). 16

These are determined by Eqs.(3.98)–(3.101),(3.113)–(3.116), where the con- 17

stants ci, di are related by the orthonomality conditions (3.112), in which the 18

bilinear form (v,u) is given by Eq.(3.107). 19

3. By comparing Eq.(3.233) with the general form (3.90), identify α , β , and f for 20

this system. This will permit you to write down Eqs.(3.155) and (3.156), in which 21

(ni,Fxt) is given by Eq.(3.157) and xt = y1(t)s1(θ)+ y2(t)s2(θ). 22

4. Equate coefficients of y21, y1y2 and y2

2 in Eqs.(3.155) and (3.156) and so obtain 3 23

first order linear ODE’s on m1(θ), m2(θ) and m3(θ), together with 3 boundary 24

conditions. 25

5. Solve these for mi(θ). 26

6. Substitute the resulting expressions for mi(θ) into Eq.(3.136) for the center man- 27

ifold. 28

7. Substitute your expression for the center manifold into the y1-y2 Eqs.(3.132). 29

Here (ni,Fxt) is given by Eq.(3.134) and xt = y1(t)s1(θ)+y2(t)s2(θ)+w(t)(θ). 30

8. Compute Q from Eq.(3.138). 31

Answer: Q =− 4π5

(3π−2)(π2+4)2 32

118 Richard Rand

3.9 Acknowledgement 1

The author wishes to thank his associates Anael Verdugo, Si Mohamed Sah, Meghan 2

Suchorsky, and Tamas Kalmar-Nagy for their help in preparing this work. 3

References 4

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